Question

Find all the units in

(a)${\mathrm{\mathbb{Z}}}_7$

(b)${\mathrm{\mathbb{Z}}}_8$

(c)${\mathrm{\mathbb{Z}}}_9$

(d)${\mathrm{\mathbb{Z}}}_{10}$

Short answer

Find all the units in

(a)$\left[1\right],\left[2\right],\left[3\right],\left[4\right],\left[5\right],\left[6\right]$

(b)$\left[1\right],\left[3\right],\left[5\right],\left[7\right]$

(c)$\left[1\right],\left[2\right],\left[4\right],\left[5\right],\left[7\right],\left[8\right]$

(d)$\left[1\right],\left[3\right],\left[7\right],\left[9\right]$

Step-by-step solution

Find the units

Make the Multiplication table as:

${\rm O}$	[0]	[1]	[2]	[3]	[4]	[5]	[6]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]
[2]	[0]	[2]	[4]	[6]	[1]	[3]	[5]
[3]	[0]	[3]	[6]	[2]	[5]	[1]	[4]
[4]	[0]	[4]	[1]	[5]	[2]	[6]	[3]
[5]	[0]	[5]	[3]	[1]	[6]	[4]	[2]
[6]	[0]	[6]	[5]	[4]	[3]	[2]	[1]

From the table, it is observed that the units are $\left[1\right],\left[2\right],\left[3\right],\left[4\right],\left[5\right],\left[6\right]$.

Find the units

Make the Multiplication table as:

$○$	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]
[2]	[0]	[2]	[4]	[6]	[0]	[2]	[4]	[6]
[3]	[0]	[3]	[6]	[1]	[4]	[7]	[2]	[5]
[4]	[0]	[4]	[0]	[4]	[0]	[4]	[0]	[4]
[5]	[0]	[5]	[2]	[7]	[4]	[1]	[6]	[3]
[6]	[0]	[6]	[4]	[2]	[0]	[6]	[4]	[2]
[7]	[0]	[7]	[6]	[5]	[4]	[3]	[2]	[1]

From the table, it is observed that the units are [1] , [3] , [5] , [7] .

Find the units

Make the Multiplication table as:

$\mathrm{O}$	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]
[2]	[0]	[2]	[4]	[6]	[8]	[1]	[3]	[5]	[7]
[3]	[0]	[3]	[6]	[0]	[3]	[6]	[0]	[3]	[6]
[4]	[0]	[4]	[8]	[3]	[7]	[2]	[6]	[1]	[5]
[5]	[0]	[5]	[1]	[6]	[2]	[7]	[3]	[8]	[4]
[6]	[0]	[6]	[3]	[0]	[6]	[3]	[0]	[6]	[3]
[7]	[0]	[7]	[5]	[3]	[1]	[8]	[6]	[4]	[2]
[8]	[0]	[8]	[7]	[6]	[5]	[4]	[3]	[2]	[1]

From the table, it is observed that the units are $\left[1\right],\left[2\right],\left[4\right],\left[5\right],\left[7\right],\left[8\right]$.

Find the units

Make the Multiplication table as:

$○$	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]
[2]	[0]	[2]	[4]	[6]	[8]	[0]	[2]	[4]	[6]	[8]
[3]	[0]	[3]	[6]	[9]	[2]	[5]	[8]	[1]	[4]	[7]
[4]	[0]	[4]	[8]	[2]	[6]	[0]	[4]	[8]	[2]	[6]
[5]	[0]	[5]	[0]	[5]	[0]	[5]	[0]	[5]	[0]	[5]
[6]	[0]	[6]	[2]	[8]	[4]	[0]	[6]	[2]	[8]	[4]
[7]	[0]	[7]	[4]	[1]	[8]	[5]	[2]	[9]	[6]	[3]
[8]	[0]	[8]	[6]	[4]	[2]	[0]	[8]	[6]	[4]	[2]
[9]	[0]	[9]	[8]	[7]	[6]	[5]	[4]	[3]	[2]	[1]

From the table, it is observed that the units are [1] ,[3] ,[7] ,[9] .